THE  TRANSITION  CURVE 

OR 

CURVE  OF  ADJUSTMENT 


BY  THE  METHOD  OF  RECTANGULAR  CO-ORDINATES 
AND  BY  DEFLECTION  ANGLES 

(POLAR  CO-ORDINATES) 


BASED  ON  THE  FRENCH  OF 

M.  NORDLING 


WITH  ADDITIONAL  PROBLEMS 
BY 


N.  B. 

M.  AA\.  SOC.  C.  E. 


(COPYRIGHT  1899  BY  N.  B.  KBLLOGG) 


SAN  FRANCISCO  : 
N.  B.  KELLOGG,  PUBLISHER,  420  CALIFORNIA  STREET 

1899 


PRESS   OF 
UPTON    BROS. 
PRINTERS    AND 
PUBLISHERS 
409    MARKET    ST. 
SAN    FRANCISCO 


PREFACE. 


The  chapter  devoted  to  the  cubic  parabola  is  derived 
from  the  method  introduced  by  M.  Nordling,  as  explained 
by  him  in  the  "Annales  des  Fonts  et  Chausse*es  "  1867. 
Translations  of  a  portion  of  this  article  have  appeared 
from  time  to  time,  but  only  so  far  as  related  to  connecting 
a  straight  line  with  a  circular  curve.  That  portion 
relating  to  connecting  circular  curves  of  different  radii, 
by  means  of  the  cubic  parabola,  has  not  appeared  in 
the  form  given  by  Nordling,  so  far  as  I  am  aware.  The 
formulae  deduced  :,  in  the  *  latter  case  are  of  general 
application  and  equally  true  for  connecting  curve  with 
curve  or  curve  with  tangent,  when  proper  values  are 
introduced  into  the  equations. 

That  of  joining  a  tangent  with  a  curve  is  a  special 
case  where  one  of  the  radii  becomes  infinitely  great. 
Some  of  the  recent  spirals,  adopted  as  curves  of  adjust- 
ment in  railroad  location,  easily  develop  from  the 
equations  of  the  cubic  parabola  by  making  the  proper 
substitutions  in  them. 

Following  the  supposition  indicated  by  M.  Nordling 
i.  e.,  considering  x  =  L  and  substituting  L  for  x  in  the 
equations  of  Chapter  I,  the  formulae  of  (61 — 70),  were 
obtained  by  the  writer  during  the  summer  of  1884. 


While  the  theory  of  the  transition  curve  as  here 
developed  is  based  upon  that  of  Froude  or  Nordling,  it 
is  believed  that  some  of  the  problems  and  tables,  used  in 
its  application,  have  not  appeared  before. 

The  figures  used  to  illustrate  the  transition  curve  repre- 
sent spirals,  as  no  interest  attaches  to  the  cubic  parabola  in 
this  connection  after  it  reaches  its  minimum  radius,  which 
occurs  when  its  central  angle  becomes  24°o6' ;  up  to  that 
point  the  figures  serve  as  well  for  the  one  as  the  other. 
OQ  represents  the  relative  position  of  the  cubic  parabola 
to  that  of  the  spiral  having  the  same  origin.  The 
examples  given  for  illustration  throughout  Chapters  I 
and  II,  are  based  upon  the  same  data,  so  that  the 
results  by  the  different  methods,  may  be  readily  com- 
pared ;  within  moderate  limits  it  will  be  observed  the 
differences  are  small.  A  few  simple  equations  of  the 
calculus  are  used  to  derive  the  formulae,  but  a  knowledge 
of  it  is  not  necessary  for  their  application. 

The  larger  type  may  be  read  independently  of  the 
smaller  type.  The  latter  may  be  used  when  the  curve  is 
extended  beyond  ordinary  limits,  and  greater  accuracy 
in  results  is  sought. 

By  the  "length  of  the  inclined  plane"  (really  a 
warped  surface)  the  "  run  off"  is  to  be  understood,  /'.  e. 
the  length  of  the  transition  curve. 

The  more  important  formulae  are  in  full  faced  type. 
In   the   arrangement  of  the  text,  I  am   indebted   to 
Prof.  H.  I.  Randall  for  valuable  suggestions. 

N.  B.  K. 
SAN  FRANCISCO,  1899. 


CONTENTS 


CHAPTER  I. 

THK   CUBIC  PARABOLA. 

Section.  Page 

1  Fundamental  Equation  of  the  curve •'! 

2  Radius  of  Curvature 5 

2  Equation  of  the  Curve  by  Rectangular  Co-ordinates..   .  6 

3  Compound  Curve 7 

3  Central  Angles 7 

4  Offset  Distance 8 

5  Computing  of  Ordinates 12 

6  Method  of  Deflection  Angles 12 

7  To  determine  Ordinates  from  Chord 14 

8  To  determine  Chord  Lengths 15 

9  Length  of  Cubic  Parabola 15 


CHAPTER  II. 

THE   SPIRAL. 

10        Equation  of  the  Curve 17 

10        Equation  for  Compound  Curves 17 

10        Central  Angle  for  Compound  Curves 18 

10        Offset  Distance 19 

10  Froude's  Formula 20 

11  Remarks  on  Preparation  of  Tables 20 

12-13      Remarks  on  Theory  of  Deflection  Angles 21 

14  Method  by  Deflection  Angles 23 

15  Verification  of  Certain  Equations 24 

16  Difference  in  Length  of  Spiral  and  Circular  Arcs  sub- 

tending same  angle 26 

17  Distance    between     Centers    and    Transformation  to 

Rectangular  Co-ordinates 28 

18  Froude's  Formula 34 

18  Passing  to  Equations  of  Cubic  Parabola 34 

19  Remarks  on  Values  of  r,  ft,  and  <£ 35 

20  Remarks  on  Use  of  Transition  Curves 35 

21  Problem  I.    Semi-Tangents 36 

21  External  Secants 38 

22  Problem  II.     Location  of  Offset  /. 39 

23  Problem  III.     Compound  Curves 41 

24  Problem  IV.    Tangents  to  Two  Curves 44 

Explanation  of  Tables 46 

25  Laying  out  by  Rectangular  Co-ordinates 46 

26  Laying  out  by  Deflection  Angles 47 

Tables  by  Rectangular  Co-ordinates 50 

Tables  by  Deflections 55 

Transition  Curve  in  old  track 60 


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CHAPTER  I. 


THEORY  CF  THE  CUBIC  PARABOLA,  OR  TRANSITION 

CURVE,  AS  APPLIED  TO  THE  ALIGNMENT 

OF   RAILROADS. 


\  1.  Let  OG,  and  A,B,  (Fig.  i)  be  respectively,  a 
tangent  and  ciicular  cuive  to  be  united  by  a  cubic 
parabola.  Let  pt  (the  length  of  the  inclined  plane)  be 
the  distance  measured  on  OG,  and  sensibly  equal  to 
OB,—  L,\  rt  the  radius  of  A4Bt. 


From  mechanics,  the  expression  for  the  centrifugal 
force  of  a  body  moving  in  a  curvilinear  path  is  : 


*/= 


32.  ir 


from  which  we  determine  the  superelevation 


of  the  outer  rail  to  produce  a  horizontal  force  due  to  gravity 


4  TRANSITION   CURVE. 

to  neutralize  this  centrifugal  force,  this  superelevation  is: 


32 


in  which  r.  =  the  radius  of  curvature  of  the  path 

in  feet  ;  v  -  the  velocity  of  the  moving  body  in  feet  per 
second,  and  g  =  the  gauge  of  the  track,  or  width  of  the 
path  in  feet.  32.2  =  acceleration  of  gravity  in  feet  per 
second.  The  transverse  inclination  of  the  track  (or 
"cant")  to  produce  the  horizontal  force  due  to  gravity* 
must  be  acquired  gradually,  and  if  we  take  a  distance 

"/"  to  rise  one  unit  in  height,  then  -4-  will  denote   the 


*I,et  w  =  the  weight  of  a  moving  body  con- 
strained to  move  in  a  curvilinear  path  f—  the 
centrifjgal  force  developed  by  the  weight 
revolving  about  a  center  with  a  radius  r.  Then 
the  expression  by  mechanics  for  the  centrifugal 
force  is 


Conceive  the  center  of  gravity  of  the  weight  to  be  moved  in  a 
direction  contrary  and  parallel  to  that  of  the  centrifugal  force — 
while  its  center  of  support  remains  the  same— until  a  horizontal 
c  mponent  of  the  force  due  to  gravity  is  developed  equal  and 
opposed  to  the  centrifugal  force.  Denote  this  component  by  w,\  then 

we 
by  similar  triangles  we  have  w  :  w,  =  g :  e\  whence  w,  = now 

by  the  conditions  imposed 


we 

.,-/...—= 


close  approximation  and  the  one  generally  used. 


stant.  The  error,  however,  is  small  in  an  cases  wnere  it  is  used  tor 
superelevation.  It  may  not  be  out  of  place  to  remark  here,  that 
it  is  the  usual  custom  in  fixing  superelevation,  to  depress  the  inner 
rail  below  the  grade  of  the  center  line  of  the  track  %e  and  elevate 


THE    CUBIC  PARABOLA.  5 

the  outar  rail  above  the  grade  line  %et  thus  preserving  the  center 
line  at  grade. 

A  more  nearly  correct  value  for  g  is 


The  resultant  of  w  and  wt  due  to  the  effect  of  gravity  = 

which    is    the    pressure  normal  to  the  plane  of  the    path  when 

centrifugal  force  is  developed. 

rise  in  a  unit  of  distance   and  for   any  distance   p  the 
rise  will  be  -?-  =    the  superelevation  necessary  to  pro- 

duce a  horizontal  force  due  to  gravity,  equal  and  opposed 
to  the  centrifugal  force,  hence  : 


JL 


whence 


,, 
32.2  r,  32.2 


The  second  member  of  the  equation,  for  assumed 
values  of  v  and  /,  is  constant,  and  is  placed  equal  to  P, 
hence  : 

r,p,=  P;         r,=  -|~  (3) 

\  2.    From  the  calculus  the  expression  for  the  radius  of 

ds* 

curvature  is  ;  r.  =  -77  —  -yr—  -  ,  in  which  ds  is  the  dirlerential 

dp,  d*yl 

of  OR,  and  dp,  the  differential  of  OGt  (or  S£?t).  Fig.  2;  d*y 
the  second  differential  of  BtGt  with  respect  \o  pt\  now  as 
we  have  assumed  OGt  sensibl)^  equal  to  OBn  therefore 
we  may  assume  ds  =  dpt,  which  gives,  thereby  : 

di)  3  P  d*v        6 

r'  =  j/,%:  =  p,  inverting  the  terms  dp?  =  p'  (4) 

whence  • 

ptdp,  . 

V'  i 


TRANSITION  CURVE. 


whence  : 


t   -   .        *• 
<*y,  =          J^»  integrating,  yt  = 


(6) 


in  which  _y,  and  pt  are  co-ordinates  of  the  point  Bt.  By  the 
same  process  we  may  find  for  the  point  Blt  of  the  parabola 
at  which  the  radius  of  the  circle  is  rllt  being  also  the 
radius  of  the  cubic  parabjla  at  the  same  point  and  co- 
incident with  it, 

i>  3 
>0*  =  £75*   which  is  the  equation  of  a  cubic  parabola.  (7) 

In  general  fop  any  point  of  the  cubic  parabola 
whose  co-ordinates  are  x  and  y  we  will  have : 


y      6P 


(3) 


EXAMPLE  1.    Given  x  —  150;   p  =  171900  to  find  the  ordinatejy 
by  (8)  jr  =         =     -  -P  =  3.27. 


\  3.     From    the    relation    established    in   equation 

(3),  we  have : 

P  P 

ptl  m'.J£t     .      A  =  —•  (9) 

' 


THE   CUBIC   PARABOLA. 

If  now  we  denote  in  Fig.  2 


(10) 
*«fi 

then 

/>=to;  (II) 

r—rtt 

which  substituted  in  the  values  for  pt  and  ^,,  give 

.          ^^     _   ^    .     ^  =    ^r-_  =  _fc_.(i2) 

~  —  '  ~~  — 


By(5)  ;-tan..-!  tan^;       (13) 

and  since  the  angles  are  small,  the  arcs  may  be  taken  for 
the  tangents  of  the  angles,  thus  : 

<^/_     _  P,2  dyn_      __P//2  /ryl\ 


EXAMPLE  2.     Given  #„  =  pn  =  150  ;     T3  =  171900  to  find  the 

P     2 

angle  a;<  subtended  by  Ln:     tan  a/y  =  -r—,  (supposed  =  arc  a.lt) 


Substituting  the  values  of  ^   and  P  in  terms  of  f, 
r,  and  rtt  we  have  : 

a<=       *r"        .          ..»—  i  _  ;  (15) 

ir^t-rj  2^,^-rJ 

subtracting 


a  _a  =  .-    = 

2r,  rw  (rt—rlt)  irtru  (r,—rlt) 


whence 


TRANSITION   CURVE. 


EXAMPLE  3.     Given  x  =  p  =  150  ;     r,  =  2865  ;     rtt  —  818.8,  to 
find  by  (17): 

ay/  -  a,  =  J3,,  +  ft  =  0  -  t  (1  +  1)  =  75  (.00122  +  .00035)  - 
•    >  *j         rf/ 
75  X  .00157 

a//  —  a,  =  ptl  -j-  j8/  =  <J>  =  arc  .11775  =  6°  44'  -j- 


pt  =  A,D,Bt;  ^ll  =  AtlDtlBll;  if  now  we  make  alternately 
rt  and  r,,=  co  and,  at  the  same  time,  make  /3,=  o  and 
/3,,  =  o,  we  have  : 


whence 

fin  rn  =  ft i  **/  —  y!tP  >    P  F  ~\~fi    V    ~P  /  ( 19) 

*The  values  of  p,  $„  ay  an  and  <|>  are  expressed  in  arc  at 
radius  =  1.  The  arc  of  a  circle  which  is  equal  to  the  radius  in 
length  =  57.295.  The  arc  of  1  degree  at  radius  =  1  is  .01745  +. 
Hence  the  expression  for  any  number  of  degrees  in  a  given 

arc  A 
circular  arc  is  A  =  —..,.,..  ,  • 


\  4.  To  find  the  distance  A.A,,^  between  the 
two  circular  curves  measured  on  a  common  normal  passing 
tnrough  their  centers;  i.  e.,  the  least  distance  apart  of 
the  circular  curves  to  be  joined  by  the  cubic  parabola: 
Referring  the  co-ordinates  of  the  circle  with  radius  rt  to 
the  same  origin  as  the  cubic  parabola  in  which  mt=.NtHt^ 
we  have  : 

;  (ao) 


squaring  the  last  term  of  the  second  member  then, 

iy^i.+ir.m.+mt+r*;  (21) 


suppressing  all  terms  of  yt  and  mt  in  which  rt  does  not 
enter  as  a  factor,  since  y,  and  m,  are  small  compared  with 


THE   CUBIC   PARABOLA.  9 

rlt  (the  opposition  of  signs  will  still  further  diminish  the 
error)  and  factoiing  the  remaining  terms,  we  have  : 

2r,  (ft—m,)  =  (p.—  l,}2 ;  (22) 

in   which  I,  is  the  abscissa  of  the  center  of  the  circle 
and  —  y^p, 

^'~  zr,  '     &rf      6r,' 

whence 

,or  in  general,  m  = 


'     br,     8r,       48^,       24^ 

EXAMPLE  4.    Given  x  =  150;   r  =  1146  to  find  the  length  of  the 
normal  "/«"  common  to  and  between  the  tangent  and  circular  curve 
to  be  joined  by  a  cubic  parabola :     By  the  general  equation 
*3          (150)2    _  22500 
24r       24XH46       27504 

In  the  same  way  we  may  show  that 

[  which  mit  =  Ntl  Hlt  (25) 

At\f  n 

To  prove  that  the  cubic  parabola  and  circle  have  a  common 
tangent  at/^y,/  i.  e.,  are  tangent  to  each  other,  we  differentiate 
their  respective  equations  :  for  the  circle, 


%-  -     =      =        -  (since  /,  =  XP.)  =  tan  a, ;  (26) 

off  rt  trt 

for  the  cubic  parabola, 

$7=^-  =  ^-(since/J  =  ^')  =  tan"':  (27) 

hence  the  tangents  coincide. 

Substituting  the  value  of  ft  in  terms  of  p 
S>2  r  2  S>2  r  2 

m,=        P(     " rr    also   mlt= p     '        •>     (28) 

24^  (rt  —  rtt)z  24;  ,Ar/~~r//) 

we  now  have  mt  and  mtl  in  known  terms,  and  by  placing 
their  values  in  the  equations  for  yt  and  ytt  in  which  the 


10  TRANSITION   CURVE. 

ordinates  for  the  parabola  and  circles  of  radii  rf  and  rtt 
have  the  same  abscissa  x  we  have  for  the  parabola, 


,       -  ,  „ 

for    the    circles    yt  =  ^f+~-  -  ;    y,,=  mu-\  --  —  ^ 

17  t  27  ^ 

substituting  in  equation  for  yt    P   =      J_  "  . 


_      t 


r,  (rt—r~ft) 

)x-trJ*  (     , 

2 


by  the  same  process 

y  =^">'<'+3C2(»-<-rjT-/r,]«.  (     } 

~      2 


(r,-^^  _3  [2  (n-rJr-^T+^r,,'.  (     v 


,_-r-r,,    x     .tr.-rx-r.  ,     . 

- 


If   now    we  substitute    for  x;   P,-\-lAp-plt  —  %P  and 
p  —  P  t^n   jn  equations  f^  the  values  of  y,  yt  and  yM 


__     _ 

-         '    " 


r  2+3     2(^-0  —  r/"r/l-^ 

-  (35) 


f  UNIVERSITY  \ 

^*^*W*#£*S' 

THE   CUBIC   PARABfttA.  II 


= 


multiply  and  divide  the  second  member  of  the  equation 
by  2rtl 


2-r/// 
-rJ^ 

^//3;  substituting,  cancel- 
ling and  factoring, 

y-y  -  .  P^,-^_  =  /2(^-^.  (39) 

"    ^-48r/r/Xr/-rJ2          48r,rw 

By  similar  reduction, 


which  is  exactly  the  same  value  obtained  for  y  —  yn 
wh;ch  shows  Uiat  the  cubic  parabola  bisects  /(or  m) 
and  each  expression  equals  l/2f  whence  adding  the 

Values  yi—y+y—y^yu—y,  ; 


v_v_ 
7*     ^- 


24 


if  the  central  angle  of  the  cubic  parabola  be  small,  not 
exceeding  say  10°. 

EXAMPLE  5.    Given  x  =  p  =  150,    r,,=  818.8,    r,=  2865  to  find  the 
offset  by  (41), 

f=  -£i  (-1 1-)  =A,A,^(^-  (.00122 -.00035)  = 


22500X_00087=0_815_ 


12  TRANSITION   CURVE. 

X* 

$   5.      From    the  general  equation  y  —  — ^  we  see 

that  the  ordi nates  vary  as  the  cube  of  their  abscissas. 
Having  computed  the  value  of  any  ordinate  y,  we  have 
for  any  other  ordinate  y0\ 


hence    ^ 


(43) 


EXAMPLE   6.     Given  x  =  120,     x0=  150,      ^=1.67  to  find   any 
other  ordinate  y0  by  (42;, 


y0= 


=  (1-25)  3X1.67  =  3.27. 


This    will   facilitate  the  computation  of  any  ordinate 
following  the  first. 

It  is  to  be  noted  that  all  of  the  above  equations 
are  approximately  true  only  for  small  central  angles. 

$6.  To  determine  the  method  whereby  the 
cubic  parabola  may  be  laid  out  by  deflection 
angles,  and  the  length  of  the  corresponding  chords, 


we  will  assume  equal  increments  of  x  of  finite  length, 
(though  they  may  be  taken  unequal)  from  which  the 
corresponding  chords  (of  unequal  length)  may  be 
computed  as  well  as  the  deflection  angles. 


THE   CUBIC  PARABOLA.  13 

By  a  previous  equation 


EXAMPLE  7.    Given  x  =  150  ;  y  =  3.26,  to  find  a  by  (44); 

tan  a  =  -^—  =  ^^  —  .0652  =  tan  3°  44' 

which  shows  that  the  tangent  of  the  angle  which  a 
tangent  to  the  curve  makes  with  the  axis  of  x  equals  the 
ordinate  divided  by  */£  the  abscissa  of  the  same  point. 
If,  therefore,  we  have  as  in  Fig.  3,  the  cubic  parabola 
ABCD, 

^-^\.2,nBAB,\  (45) 

and 

^=tana0;  (46) 

then 

tan  BAB,  =  — —  =  tan  70 ;  (47) 

tan  TI  =  yi"V°;  (48) 

X-l       Xo 

tan72=  x'lx'  ;  (49) 

707i72  are  t*16  angles  the  respective  chords  make  with 
the  axis  of  X:  72~7i  is  the  angle  which  chord  CD 
makes  with  BC.  By  a  similar  process  the  angle  which 
chord  AC  makes  with  CD  can  be  computed.  For  small 
central  angles  the  tangents  of  the  angles  may  be 
assumed  equal  the  arcs  so  that 

^'f1   -"^"T^  =  72-7i  =  A  (50) 


the  exterior  angles  between  the  chords. 


14  TRANSITION   CURVE. 

The  several  angles  can  be  computed  and  tabulated, 
to  any  number  which  is  likely  to  be  needed,  to  conform 
to  any  system  of  "change  points"  determined  upon 
aft^r  X0  y0  &c.  have  been  computed  for  the  particular 
transition  curve  where  value  of  P  has  been  fixed  in 
conformity  with  the  character  of  the  alignment.  The 
foregoing  method  for  computing  deflection  angles  is 
equally  applicable  to  the  spiral. 


S.  Given  X0  —  fiO  ;  #i=120;  X2  =  209.65;  y0  =  .21  ; 
yi=\  f>7  ;  ^2=8.91,  to  find  the  deflection  angles  for  laying  out  a  cubic 
parabola  which  successive  chords  make  with  each  other 

-^2-  =  tan  y0  =-  -^-  =  .0035  =--  tan  0°  12'  ; 
..-.-  urn  T1,  L-.  0245=,  an  1-24-  ; 


yi  -  yo  =  1°  21'  -  0°  12'  =  1°  12'  ; 

Y2  _  yi  =  4°  37'  —  1°  24'  =  3°  13;  ; 

For  long  chord;  -g-  =  tan  y  =  A5L  _  2*  27,  . 

'iio>ii  tt 

comparing  the  value  of  a/y  and  y  we  see  y  =  ••—  ~. 

o 

2  7.    To    determine   the   ordinates  o,   o1   &c. 
From  any  chord  as  Ic,  let  aB  —  o,  B^b  =  s,  ABl  =  iC0, 

B^B  =  /0,    M^x  7.     Then    from    Fig.    3,  -  —  =  tan  7, 

^o 

s  =  x0  tan  7,  J  -  y0=  ^0  tan  7  -J0.  ~    -  =  cosy,o=(s  -yc) 

s     So 

cos  7,  or  since  s  =  <r0  tan  7, 

o  =  (X0tan7-y0)eos7  =  X0sin7-y0  cos7;   (sO 

For  the  distances  Aa  &c.,  ^^  =  AlBl^  secy  -alt,  tan  7 
=  a*0,  sec  7  -  d?  tan  7.  In  the  same  way  we  may  obtain 
o±  and  Aa1  to  determine  the  point  B;  C  &c.  by 
measurement  alone.:  First  compute  and  lay  off 


THE   CUBIC   PARABOLA.  15 

the  distances  Aa,  Aa^  &c.,  then  lay  off  AB  and  aB 
simultaneous!};  next  BC  and  a^C  &c.;  when  7  is  small 
the  distances  bB  and  b^C  may  be  used,  distances  Ab  and 
Abl  being  computed  and  laid  off  first. 

|  8.  As  a  check  it  will  be  an  advantage  to  compute 
the  length  of  the  long  cord  as  well  as  the  angle  it  makes 
with  the  axis  of  x,  thus: 

tan  DAD!  =  ^^  =  ^-,  (52) 

AJJ  i        Jin 

in  which  (ri)  equals  the  number  of  increments  or  stations 
between  A  and  D.  L,et  the  length  of  ai.y  chord 
be  CL ,  C2  or  c3  ;  then 

c2  =  (x2  m*#i)  sec  72 ;      c±  =  (x±  -  ^*3)  sec  74  ;     (53) 

EXAMPLE  9.  Given  X2  =  20X65 ;  Xi  -  120  ;  y2  =  4°  37',  to  find 
the  length  of  chord,  c2  =  (X2  —  x^  secy2  =  89.65  X  1.003J  =  SU.93. 
For  the  length  of  long  chord  (from  origin),  X2  —  209.65 ; 
V  =  2°  27';  x2  sec  y  =  209.65  X  1.009  =  209.84. 

If  the  central  angle  is  considerable,  the  error  arising 
from  considering  the  arcs  equal  the  tangents  of  the  angle, 
will  be  too  great,  and  the  expression  tan  a,  tan  7  &c. 
must  be  adhered  to.  It  will  be  observed  that  the  field 
work  will  often  be  facilitated  by  laying  out  the  parabola 
by  ordinates  and  calculating  its  length  to  preserve 
correct  measurements  along  the  line. 

2  9.  For  the  rectification  of  curves :  by  the 
calculus  we  have 

ds2  =  dx*  +  dy-  =  dx2  +  —^  dxz,  (54) 

and  since 

Htx   ~  ~2P"      ~d~x^  ~  Z/72'  ^' 


16  TRANSITION  CURVE. 

substituting  and  factoring 


developing  the  radical  by  the  binomial  formulae 


Integrating    the    last    equations    between    the     limits 
Ln  and  L,—  L,  xtl  and  x,  we  have 


(58) 

since  /*  =  rw  ^  =  r,  ^,, 


in  which  L  =  the  length  of  the  cubic  parabola  uniting 
the  curves  with  radii  rt  and  rlt.  If  rt  —  co,  Lt  =  o  and  L 
unites  the  curve  with  radius  rn  with  a  tangent  and  the 
second  parenthesis  =  o,  whence 

L  =  xit(i  + 


—  irTT" 

I33I2/*6 


EXAMPLE  10.  Gix^en  x,,  —  209.65  ;  x,  =  60  ;  ry/  =  818.8  ; 
r,  =  2865,  to  find  the  length  of  the  parabolic  arc  connecting  the 
circular  curves  with  radii  r,,  and  rn  by  59  : 


-  *» 


+ 


60 


CHAPTER  II. 


If  in  equations  3,  5,  8,  15,  17,  28,  39,  41  obtained  by 
the  foregoing  process  we  write  L  for  p  (or  x}  since  they 
have  been  assumed  to  be  equal  for  small  central  angles, 
the  resulting  equations  will  give  a  much  nearer  approxi- 
mation to  the  true  transition  curve  for  a  central  angle  up 
to  15°,  which  is  rarely  exceeded,  thus  : 


-L_.:$.  Fig  2 


I J  •  I         f^  f%~'         /V 


,=^r      (6l) 


Lp=P; 


which   is  the   ideal   equation   of   all   transition   curves. 


i8  TRANSITION  CURVE. 


EXAMPLF.  11.  Given  P  -  171900;  r,,  =  818.8  ;  r,  =  2865,  to 
find  the  length  of  the  transition  curve  uniting  the  circular  curves 
with  radii  r,,  and  /•,,  by  (62) 


L  -  P  (— — ")  =  171900  (  00122  -  .0035)  =  171900  X  .00087  =  150. 

V   f //  ft     / 

If  curves  are  reversed  change  the  sign  of—  from  —  to  +;  then 
L  =  L,,  +  L,  =  171900  X  .00157  =  210  +  60  =  270. 

^rr;       (63) 


L3  dx 

=  «-=:;  an(i  since  — p  =  COS  a, 
D  i  Q  Ju 


Lrlt 

— ^ — .:  (65) 


subtracting  and   reducing  to  a   common   denominator 
and  factoring : 


EXAMPLE  12.      Given  /.=  150  ;     r,,  -  818.8  ;     r,  -  2865,  to  find 
the  angle  subtended  by  the  arc  of  the  spiral,  by  (66): 

a/;  —  a,  =  £;/  -f  /3y  =  -^  ( 1 }  —  arc  <^  = 

75  X  .00157  =  .11175    </>°  =  6°  45'. 

For  r,  =  oo  -1-  =  0,  /3;  =  0,  arc  <£  =  -—  =  arc  £„=  .0915,  ptl°=  5°  15'. 
rt  I  r,, 

For  rn  =  »  —  =  0,  /3/y  =  0,  arc</»  =  -^-  =  arc  0,  =.02625,  /3,°=  1°  30'. 
rn  ir, 


Ntt  Htt  =  mtl  =  -----        1 
"     "        "      24^^- 

/  2r    2 


(.67  ) 


THE   SPIRAL.  19 

y  ->»-  -^8  n?1  ='«  -'=  */;       (68) 

—  the  ordinate   of  the   middle   point  of  L. 


in  which  L  =  P  {  -^—  ^ )  = 


EXAMPLE  13.    Given  L  =  150;    r/y  =  818,8;    r,  =  2865,  to  find 


/  =  --  X  .00087  -  .815 


For  reverse  curves  the  perpendicular  distance  between  the 
tangents  to  the  circular  curves,  produced,  which  are  parallel  to  a 
tangent  to  the  common  origin  of  the  transition  curves  is  expressed  by 


In  this  and  all  examples  given  on  the  above  data,  it  is  to  be 
observed  that  the  value  (d)  which  will  appear  in  subsequent 
investigation  is  inappreciable  and  need  not  be  regarded,  also  that  the 
cubic  parabola  and  spiral  are  practically  the  same  and  the 
formulas  of  either  may  be  used  indiscriminately.  Table  (2)  may  be 
regarded  without  serious  error  as  adapted  to  either  case. 


If  the  curves  rn  rt  are  reversed,  use  the  +  sign;  if  in 
the  same  direction,  use  the  -  sign  in  equation  (69), 


TRANSITION  CURVE. 
Solving  (69)  for  y^L  we  have 


which  is  Fronde's  curve  of  adjustment.  The  above 
equations  may  be  obtained  by  an  independent  method,  as 
will  be  shown  further  on,  where  the  values  of  x,  y  and  /, 
respectively  will  appear  as  the  first  term  of  a  series.  In 
all  the  above  equations  if  r,  be  made  equal  to  infinity, 
the  terms  in  which  it  appears  become  =  o,  the  terms 
containing  it  disappear,  and  the  resulting  equations  are 
for  uniting  a  circular  curve,  with  radius  rin  with  a  tangent, 
by  means  of  the  transition  curve.  If  the  sign  of  r,  be 
changed  and  Ltl  and  Z,,  be  substituted  for  L  in  connection 
with  rlt  and  rt,  the  equations  will  be  those  for  uniting 
curves  reversed,  by  means  of  the  transition  curve, 
so  that  (61-69)  inclusive,  are  general  equations  for 
the  transition  spirals  and  are  sufficiently  accurate  when 
they  are  used  for  computing  spirals  for  transition  pur- 
poses only. 


?  11.     Tables  I  and  2  are   prepared   by  the  use  of 

equations  62,  63,  64,  66  and  69,  in  which is  made  zero, 

except  where  the  value  of  x  is  considerable,  when  115  is 
used. 

A  table  should  be  computed   consistent  with  the 
character  of  the  alignment  to  which  it  is  applied,  of 
which  table  2  may  be  taken  as  an  example  where  — 
in  the  formulas  has  been  made  =  zero. 

It  is  to  be  observed  that  the  superelevation  of  the 
outer  rail,  in  the  use  of  the   transition  curve,  may  be 


THE  SPIRAI,.  21 

made  greater  or  less  than  that  which  has  been  assumed 
in  computing  the  tables  ;  the  only  effect  it  will  have  is  to 
diminish  or  increase  the  assumed  value  of  "z ",  which  is 
equivalent  to  increasing  or  diminishing  the  velocity, 
since  i  and  v  may  be  made  variables  in  the  constant  Pt 
i.  e.,  it  makes  the  rate  of  rise  of  the  outer  rail  to  effect 
superelevation  a  little  greater  or  less.  It  is,  however, 
best  to  introduce  the  average  velocity  of  the  express  or 
fast  passenger  trains  in  constructing  the  tables.  Where 
the  location  is  so  constrained  that  the  PT's  and  PC's 
of  the  circular  curves  are  quite  close  together,  it  may 
be  necessary  to  give  "2 "  a  smaller  value  than  would  be 
otherwise  desirable.  A  value  of  300  or  400  is  sufficient 
for  adjustment,  and  good  results  may  be  obtained  with 
a  value  of  200  when  the  radius  is  not  greater  than  573 
feet.  Equations  (62-69)  contain  all  the  elements  necessary 
to  determine  a  suitable  curve  if  intended  only  for  the 
purpose  of  adjustment,  and  on  account  of  their  simplicity 
are  recommended  for  ordinary  use.  The  latter  remark 
presupposes  that  Problem  III  will  be  used  when  the 
distance  assunder  (or  /)  of  the  two  circular  curves  to  be 
united  is  much  in  excess  of  that  required  for  a  transition 
curve  with  the  fixed  value  of  P. 


\  12.  If  at  the  point  Bt  (see  Fig.  2)  we  imagine  r,  to 
increase  till  it  becomes  equal  to  infinity,  the  curvature  of 
B,Af=  o  and  the  arc  BtA,  will  be  a  straight  line  still 
preserving  its  tangency  to  the  transition  curve.  The 
curvature  at  Btt  will  diminish  to  the  same  extent,  i.  e., 
the  difference  between  curvature  at  Bt  and  Bu  will  be  the 

L'2 
same  as  when  j8,  =  o,  and  £„  =  — — .     The  ordinates  x  and  y 

can  be  computed  and  laid  off  from  the  new  tangent  as  axis 
of  abscissa  with  Bt  as  origin  the  same  as  if  from  O.  If  we 
now  conceive  this  new  axis  of  x  to  be  curved  to  a  radius 
rt  the  curvature  of  the  transition  curve  at  any  point  will 


22  TRANSITION   CURVK. 

be  increased  by  the  same  amount  and  the  ordinates  mayf 
without  serious  error,  be  laid  off  normal  to  the  arc  BtA, 
and  establish  points  of  the  transition  curve.  The  same 
reasoning  will  apply  if  r/7=coand  BltAn  becomes  a 
tangent  and  values  of  x  and  y  be  laid  off  from  it  with 
Btl  as  origin,  except  that  the  resulting  transition  curve 
would  be  convex  to  BtlAl(.  The  ordinates  would, 
however,  be  equal  to  those  of  the  corresponding  distance 
from  B.  If  ru  now  resume  its  original  length  the 
curvature  of  the  transition  curve  at  any  point  will 
equal  that  of  the  circular  curve  with  radius  r  n  minus 
the  curvature  it  had  in  a  contrary  direction  when 
rlt  was  equal  to  infinity  and  Blt  Atl  a  straight  line.  In 

determining  $„  and  r  =  -.—  ,  data  may  be  taken  from  the 
tables. 

$  13.     To  conform  to  the  case  under  consideration 


r  will  be  measured  on  AtBn  and  y  normal  to  A,fft. 
Kquations  for  x  and_y  are  equally  true  whether  the  origin 
be  taken  as  O,  Bt,  or  Btl.  To  lay  the  transition  curve  out 
by  the  expressions  for  x  and  j,  their  values  may  be  laid  out 
simultaneously  with  corresponding  equal  chord  measure- 
ments along  the  transition  curve.  It  will  be  seen  that 
the  above  values  of  x  and  y  differ  from  those  of  the 
cubic  parabola  slightly  (which  attains  its  minimum  radius 
when  the  central  angle  becomes  =  24°  6')  and  fulfill 
the  conditions  of  a  spiral  whose  curvature  and  length 
vary  inversely  as  the  radius,  and  may  become  infinitely 
great  and  the  corresponding  radius  infinitely  small  or 
equal  o.  As  this  extreme  is  never  reached  in  practice, 
its  discussion  is  beyond  the  scope  of  this  paper  and  will 
not  be  considered  here. 


THK  SPIRAI,.  23 

|  14.  The  principle  enunciated  in  the  paragraph 
preceding  the  above,  enable  us  to  prepare  a  table  of 
deflection  angles  according  to  the  following  method, 
which  may  be  more  convenient  than  that  laid  down  in 
Chapter  I. 


Fig  4- 


— x 


Refering  to  Fig.  4.  If  the  deflection  angles  from 
AX  to  any  point  be  denoted  by  d  5,  5,,,  it  will  be  found 
by  computation  that  any  angle  as 

DAX=  3,,=  4f  =  ST-  (nearly),  (71) 

in  which  rl4  —  DK. 


the  angle  which  CT,  a  tangent  common  to  the  spiral  and 
circular  curve  with  radius  rtlt  makes  with  the  chord  AD.r* 
To  establish  the  points  E,  F  arid  G  by  deflections  at  D 


24  TRANSITION  CURVE. 

from  tangent  DT  we  have,  from  the  paragraph  already 
referred  to, 

EDT=8  +  &,  (73) 

and  5  —  the  deflection  from  AX  to  Z?,  and  A  =  the 
deflection  from  DT  to  the  point  E,  for  the  circular 
curve  with  radius  rtl.  In  the  same  way 

GDT '=  «„+ A,,;  (74) 

the  angle 

DGG,,  =  25,,  +  A,,=  (a,,  -  a,)  -  (5,,+A,,).          (75) 

If  we  add  (8,,-f-A/,)  to  both  members  of  the  equation,  we 
have : 

GOT  =  a,,  -  a,  =  0  =  3  «„  +  2  A/,-  (76) 

in  which  A«=^A»  *n  which  A  =  the  deflection  for  a 
unit  length  of  DG  on  the  circular  curve  with  radius  rin 
and  L  the  units  from  D  of  any  point  laid  off  on  DG. 

The  above  values  are  approximate.  If  close  results 
are  indispensable,  the  method  indicated  in  connection 
with  Fig.  3,  in  determining  5  5,  5,,  will  have  to  be 
resorted  to. 


§  15.    As  indicated  on  page   20,    we  now  proceed  to    verify 
certain  equations  (3-11,  61-691. 

We  have  from  the  calculus: 

rda  =  dL,    L  =  £=Pr-i,  (77) 

differentiating 

Pi*—  2  ffr 

<lL  =  Pr-2dr,    rda  =  Pr~2dr,    d*  = **  Pr~*dr;    (78) 


THE  SPIRAL.  25 

«=/»•-'  *-=jiri=  27?  (79) 

hence  for  any  two  angles  an  and  a,  we  have  : 

a"  =  W^'     a'  =  2>^;  (8o) 
P            P         P(   1             1    \ 

a"  ~a'  =  ~2^r  ~  W  =  -2  V7~5  ~  Tjr  (8l) 


From  (62),  L,  —  —  •     L,,  —  -  ;    whence 


L         L    -L-P-  ~  (K*\ 

Ln  -L,-        p\  r^  -  —  ;,   -  -  -  \-v-  -  —  ;,    (83) 

substituting  in  value  for  an  —  a,,  we  have 

«//  -  *,  =  4-  (~-  ^  -^)  -  ^>  -  PI.  *  &;          (84) 

M^  +  j/^«  being  used  when  curves  are  in  the  same  direction  and 
the  —  sign  when  the  curves  are  in  a  reversed  direction.  (If  the 
—  sign  is  used  the  second  member  of  (84)  will  still  equal  a.,,  —  a/, 
in  which  a;/  is  on  the  opposite  side  of  the  axis  of  X  from  a,,  or  the 
curves  are  reversed,  L  =  Llt  +  L,,  a.,,  =  /3y/  and  a,  =  0,)  in  which 

<*>  -  ftu  +  f*i  (85) 

is  the  angle  subtended  by  the  arc  L  of  the  spiral  as  well  as  the 
sum  of  the  circular  arcs  A,  B,  +  A,,  B,,,  or  of  a  circular 

2 

arc  whose  length  is  =  L  and  radius  r  =      1  1    .*  (86) 

~  ~ 


*0  =  —  ^-  (^  --  1  --  );  L  =  r<$>,  in  which  r  =  some  length  of 
radii  that  with  a  central  angle  $  will  give  an  arc  =  L  or  L  =  r(f>t 
L  and  0  having  the  same  value  as  in  equation  84.  Then  $  =  -  = 

4-(-UJ->  —  =4-(—  +—  ),  -Wing  for  r=  /  1  '  IX 
2.  \  rtl  rt  '  r  2  ^  rlt  rt  '  (  --  1  --  ) 

^  *"//  *"/  7 
112 

--  1  --  =  --  .'.  the  reciprocal  of  r  is  an  arithmetical  mean  of 
*n  ri  r 

the  reciprocals  of  r,,  and  r,. 


26  TRANSITION   CURVE. 

If  in  equation  (84)  we  make  rn  —  oc,  and  fi,,  —  0,  then 

4r^:;fc?# 

similarly  we  may  find 

1     -B          L     - 

jj^-ft"  -r-"n,fci 

whence 

£  =  /3//r,,  T  r,j8,.  (87) 

The  angles  subtended  by  £„  and  jS,  arcs  of  spirals  in  this  equation, 
are  not  precisely  the  same  as  result  from  the  investigation  follow- 
ing. In  the  value  L,  jS,,  rn  and  ft,  r,  are  assumed  to  be  equal  as 
would  appear  from  Nordling,  and  A,/  A,,  the  shortest  distance,  to  be 
at  the  extremity  of  equal  arcs  AltBn  and  AtB,.  For  large  angles, 
this  is  not  the  case  there  is  a  difference  =  d  ;  a  radius  rt  would  not 
pass  through  the  center  of  rn  exactly.  In  LC  the  angle  is  divided 
properly.  A,  B,  <  A,,  B,,  (see  §  16.) 

If  j3;/  and  (3;  be  the  angles  subtended  by  the  circular  arcs,  at 
radius  =  1,  then 

Lc  =  |8y/r//  +  /3/ry;  (88) 

$nrn  and  ptr,  being  respectively  the  arcs  of  circles  A,,  B,,  and  A,  Bt. 

§  16.  It  has  been  assumed,  in  the  case  of  the  cubic  parabola, 
that  the  length  of  the  arc  of  the  transition  curve  does  not  differ 
sensibly  from  the  sum  of  the  lengths  of  the  arcs  of  the  two  circles 
«qual  to  each  other  with  radii  r,,  and  r,,  described  from  the  ends  of 
the  transition  curve  to  points  where  their  circumferences  approach 
nearest  each  other  and  their  tangents  are  parallel,  also  that  the 
abscissas  of  their  centers  equaled  %  the  abscissa  of  the  point  at  which 
the  circle  and  cubic  parabola  become  tangent  to  each  other  and 
have  a  common  radius.  In  case  the  central  angle  or  length  is 
considerable,  as  may  occur  in  the  use  of  the  spiral,  the  error 
arising  from  this  supposition  may  not  be  disregarded  and  the  true 
value  of  AnB,,  (see  Fig.  2)  and  A,B,  should  be  computed  in 
determining  the  length  of  arc  to  lay  off  from  An  or  A,  to  establish 
the  respective  points  Bn  and  B,.  The  arc  of  the  circle  having  the 
longer  radius, as  rn  will  always  be  the  shorter. 

The  rectangular  co-ordinates  of  the  center  of  the  circle  with 
radius  rn  as  Dn  referred  to  the  origin  O  are:  OP ',=  I,  and  H,D,=  klt 
and  of  Dn,  OH,,  =  I,,,  H,,D,,  =  &//;  whence 

~^  =  UH(«,  +&);  (89) 


THE  SPIRATv. 

llt  —  pn       r,,  sin  a,,,  I,  —  PI  —  r,  sin  a,; 

klt  =  r,,  -•-  m//(  k,  =  r/  --,-  w/; 

e 

—  ^P//  —  >*//  sin  a,,)  —  (p,  —  r,  sin  a/) 
tan  («/  :  P/)  —  —    ,        »      >        i  m   N 


or  if  we  have  the  value  of  f 


COS  (a;      0,)  = 


(P7  +  m/)  -  (P,,  4-  m;/) 


p,  -  (r,,  4-  f) 


from  either  of  these  equations 


with  ^//0  reduced  to  arc,  then 

r,,?,,  =  AnBtl 
and 


denoting  the  difference  between  L&n&AnRn  -\   A,  B,  by 
have 


Placing 


=  £-(4,;^    -  A,B,)  =  L  •-&„&„    -r,?,). 


'J-  L  -  Lc. 


27 

(90) 

(92) 

(93) 

(94) 

(95) 

(96) 

ld"  we 

(97) 
(98) 
(99) 


The  point  #„  should  be  marked  with  the  name  of  the  station  at 
/>'/  the  computed  length  of  the  transition  curve  (as  if  A,  B,  and 
Atl  Bn  were  laid  off  in  the  same  manner  as  semi-tangent  of  a  simple 
circular  curve). 


28 


TRANSITION  CURVE. 


To  determine  the  angle  /3  (second  method):  Let  Y  =  BH  B,  S, 
=  the  angle  that  the  chord  B,  B,,  makes  with  the  axis  of  X,  then 
(Fig.2a): 


Fig  2  a 


tan  Y  =  __  y/.  The  angle  which  a  tangent  to  the  curve  at  B, 
makes  with  the  chord  B,  B,,  =-  lc  is  Y  —  a.,.  The  chord  B,  B,,  = 
lc  =  \/(yt/  —  y/)2  -+.  (#/y  _  3.^2.  The  distance  Z>/7  T7=/c  cos  (Y—  a,)  - 
r,,  sin  (a,,  —  a/).  The  distance  D,V  -  r,  —  [/c  sin  (Y  —  a) 

r/y  COS  (a/;  —  a/)]. 

/C  COS  (Y  —  a,)  —  rn  sin  (a,,  —  ay> 


,  -  [/c  sin  (Y  -  ay)  -f  r,,  cos  (a/y  - 


^  />/  A,  =  ay  4-  jS,  =  a,,  -  £„, 
transposing  and  changing  signs, 

<*//  -  «/  =  ^//  +  ^/; 

§  17:    It  remains  to  determine  the  distance 
Z>,  Z>;,  =  r,  -  (r,,   ;-  /) 


(100) 


(101) 


To  do  this  we  must  first  find  an  expression  for  f  (and  also  for 
.t  and  y)  which  is  a  line  normal  to  both  circular  curves,  joining  the 
points  at  which  they  approach  nearest  each  other;  this  is  always  on  a 
line  passing  through  the  centers  of  the  two  circles;  passing  to  rectan- 
gular co-ordinates,  by  the  calculus,  we  have  since,  equation  (13): 


THE   SPIRAL.  29 

sin  <£    dL  * 

* 


dy  —  dL  sin  </>  ;  (IO3) 

rt'ic  =  rfA  cosc/>  ; 
in  which  </>  =  any  angle.     By  trigonometry, 


<h2  (t)4  </>m 

cos*  =  1  -  —  -;-  j-^      etc  ......  1-^G5;          (IO6) 


in  which  m  may  have  any  value  from  zero  to  infinity,  n  may  have 
any  value  from  one  to  infinity  and  <J>  any  value  from  zero  to  infinity. 

If  in  the  equation  </>  =  — —  ( j j  we  write  for    convenience 

( 1 \  —  -_  then  </>  —  — —  =  ~n-p\  which   value  in  equations 


for  sin  <!>  and  cos  ^  give,  by  (103): 
dy  _    .        _  L^  /  .         L  ^ 


L    /.         L^  L^  \ 

ft  V1       J2"4K^        1920  R^  iIO7) 


*It  should  be  remembered  that  L  is  an  increasing  function  of 
M  and  j,  and  for  any  value  imagined  for  dL,  dx  or  dy  undergoes  a 
corresponding  change. 

tThis  and  the  succeeding  formula  are  for  expressing  the 
trigonometrical  function  of  an  arc  in  terms  of  the  arc  itself  (see 
Chauvenet's  Trigonometry,  Chapter  XIII,  1867). 

\an  -  a,  =  -~  ( — —J  =  <f>;    make    r,  =  cc,    then    a,    —  0, 

" 


30  TRANSITION   CURVE. 

EXAMPLE  14.    Given  r,,  =  818.8;   r,  =  2865  ;    L  -  150,  to  find  </>. 


If  we  make  r,  =  &>,    —  =0,    0,  =  0  ;   we  have  approx 


150 
Si"  '»  = 


\ 
+)  =  -°91G  (1  -  -001398'  = 


-  24X(8l 
.0916  X  99866  =  .091477  .-.sin  £„  =  sin  5°  15'. 

In  the  same  way,  if  we  make  rn  —  oo,  -  —  =  0,  /3/y  =  0,  we  shall 
find  that  sin  /3,  =.02618  =  sin  1°  30'  .'.  j3/y  -f  £/  =  5°  15'  f  1°  30'=  6°  45'. 
j8/,  —  /3y  =  3°  45'  =  the  change  in  direction  of  the  tangents  to  the 
extremities  of  the  transition  curve  when  the  main  curves  are 
reversed. 


\ 

);  (I08) 


NP.r  ' 

integrating, 

^-IFV-sijir    etc ^qf);         (I09) 

or  in  terras  of  R,  since  P       A  L,  ~rr  = 

X         r,,       r. 


*  =£('-!& +«"•)  = 


EXAMPLE  15.      (iiven  r,  =  2865  ;    r/y  =  818.8;    L  =  150,  to  find 
y  by  (110): 

L*  {•&        l   \  T-,       T/2   /  1         l  \-  1 

> = -y-  Cisr^rJ  L1  -56-  (i^  -^    etc-  J 


(.00122       00035)  [1       ^—  (.00122      .00035)  2  -u  etc.] 

v     -  OD 

99^00  '  x    OOOK7 
^  =  «   >u  ^  -u    5!  ^       22500  (.00087)^    ;    etc.]  =  3.2625  (.9997]  =  3.26225. 

or  in  terms  of  </>,  since  L  —  i 
_  2    „..,/,        ^>^     .  > 


THE  SPIRAI,.  31 

similarly, 

dx  /'          L*  L* 

dL—coo^ll       _       __.. 

(1-fr2      384R. 


dx  =  dL  (l  -  g~  -r  3^5  -  46(^p7;       etc  .....  -J—-J     (  l  ,  3) 

integrating, 

fi          ^4  ^'s  '-  m  - 

- 


=       V-405PT  -'-  3456A-.        5M 
or  in  terms  of  </>,  since  L  =  2^</>, 


also 


KxA^ri'i.E  16. 

A-  =  /,  [l      —£•  (~  ----  i-)  2     etc.  (other  terms)]  = 
/         22500  X  .0000007569          \. 

~lcT 

x  =  150  (.9996  :-)  =  149.94. 

It  is  to  be  observed  that  the  above  values  of  x  and  y  do  not  differ 
appreciably  from  the  values  given  by  (8)  and  (64);  the  difference, 


32  TRANSITION   CURVE. 

however,  becomes  greater  as  L  and  (—  -     --  j  become  greater. 

x  is  laid  off  on  the  arc  of  the  circle  A,  B,  or  A,,  Bn  (with  radii 
r,  and  /•„)  as  axis  of  X  with  B,  or  B,,  as  origin,  x  being  the  abscissa 
of  y  which  is  laid  off  normal,  or  radial,  to  the  arc  A,  B,  or  A,,  B  (l. 

In  general,  by  Fig.  (2),  any  distance 

m  =  y  -  R  ver  <f>  =  y      R  (1  -  cos  f/>),  (  r  20) 

in  which  R  depends  on  --------  for  its  value,  substituting  for  cosc/> 

rn        fi 

p 

its  value,  remembering  that  /«>  =  —  =-,  writing  /  for  m  and  reduc- 

Lt 

ing,  since 

.)];      (I21) 


-fete.) 


-)]' 


1 
~ff~' 

etc.  ) ,  and  since  - 


EXAMPLE  17.  Given  r/y  =  818.8  ;  ry  =  2865  ;  L  =  150  ;  to  find 
the  offset  /.  1st,  when  the  circular  curves  r,,  and  r,  turn  in  the 
same  direction,  by  (125): 

,_  L~  (  l     l  \  fi   AIM     1  VI  - 

J  ""2T  V77      ^"^  L1       112   V  ry/       f,  /  J 


X  .00087     l--  .  .815 


THE   SPIRAL.  33 

'2nd,  when  the  circular  curves  turn  in  opposite  direction  or  are 
reversed.  Given  r,,  818.8  ;  r,  -  2865  ;  L,,  =  210  ;  L,  =  60  ;  to  find 
the  offset  between  the  tangents  to  the  circular  curves  which  are 
parallel  to  the  tangent  to  the  transition  curve  which  passes  through 
its  origin  (Fig.  9). 


-  . 

.t     J>  -  24  |_V  rn  i    ri  )       112  \r,,3  rr,s  / 
-[(53.86  +  1.25)       (.0000007)]; 

=  /„  +/  =  ^-  =  2.29.     If  LA  =  L,  ,  then 

-  Li     l          ~     L'    l     l 


If  now  we  substitute  for  /  its  value  in  the  equation  for  D,  L)n 
we  have  for  the  distance  between  the  centers  of  the  two  curves, 

•  D,  D,,  =  r,       [r,,  ~£(±-—L.)  1  -  etc.],        (la6) 

EXAMPLE  18.  Given  L  -  150  ;  r,  =  2865  ;  r,  =  818.8  ;  to  find 
the  distance  between  the  centers  of  curves  with  radii  rn  and  r,. 
1st,  if  curves  turn  in  the  same  direction,  (Fig.  2): 


2865  -   [818.8      ^|^  X  .0008?]  ; 
D,,  D,  -  2865  -  [818.8  +  .815]  =  2865      819.615  =  2045.385. 

2nd,  if  curves  are  reversed  (or  turn  in  a  contrary  direction)  (Fig.  9), 
then:     If  L,,  =  210  ;   /,,  =  60  ;    r,  =  2865  ;    r,  =  818.8: 


X  sec  A'7/  Z?y  Z);//  =  [r,       (/•„      y/y      //)]  sec  K,,Dn  Dnl  ; 
^  /),,,  =  [2865       (818.8      2.29)  j  sec  Kn  D,  Dn  =  8686.09  sec  A',,  /^  />7//; 
tan  Kn  D,  D,,,  -  — 


*  Also  D,  Dn^  D,  V  sec  £,  (by  Fig.  2a^  and  /     r--  (r,,      I),  V  sec  ^/) 


34  TRANSITION  CURVE. 


$18.     If,  in  the  foregoing  values  of  y,  x,  —  —,  and  /  we  omit 
all  terms  in  the  bracket  after  the  first,  we  have 


x  =  L  =  X\l+-j^-  (I28) 

dy_^L^=  _L_  /  J 1_\  =  ^^ 

J.r         2/>  2     Vr//        r/  /  ~     Ln  <?'  (129) 

/=~lr  (77^  TT)'  (J3o) 

the  same  as  (>r,  H-oml  <tl)  or  from  which  latter  equation  we  find: 


identical  with  the  Froud's  curve  of  adjustment,  as  indicated  by 
Prof.  Rankine  (Civil  ^Engineering,  Edition  1863;.  If  in  equations 
(127,  128,  12!),  130),  we  write  p  or  .rfor  /,,  we  again  have  the  equations 
for  a  cubic  parabola  the  same  as  in  Chapter  I. 

If    the    transition    curve    embraces    a    large    central     angle 
equations  (110,  119  and  125)  should  be  used. 

y>f—  -  70-  ( ),   and  if  in  the  value  for    r  we  write 

18    V  rn        r,  / 

L  for  L  and  cube  it,  then 


6P 

Hence  the  ordinate  y      y^f  is  at  the  middle  point  of  the  transition 
curve. 

Many  of  the  formula;  of  [§  15-17],  while  more  nearly  correct 
are    too    extended    for   field   computation.       Tables   3,  4,   etc.   are 


THE  SPIRAL.  35 

prepared  by  the  use  of  them,  and  similar  ones  should  be  resorted  to 
for  operation  in  the  field  when  those  deduced  by  [§10-11]  are  not 
close  enough. 


19.       If   /-/  =  oo,  /3,  =  0,  a,  =  0.       A,  B,  is  a  straight  line  = 
/ -n  Sin  pfi  -  .r//  _  t-n  sin  0  =  .r,/  —  '"//  sin  a//  =  OHU.  (Fig.  2.) 


2  20,  The  transition  curve  should  be  no  longer 
than  required  for  purposes  of  adjustment,  considering 
the  highest  speed  that  can  be  safely  attained  upon  the 
circular  portions  of  the  alignment  to  which  it  is  applied. 
The  longer  it  is  the  more  it  increases  the  perplexities  of 
the  trackman,  who  is  unable  to  maintain  the 
transition  curve  in  its  proper  place  wholly  "by  eye" 
or  by  their  usual  methods.  The  aid  of  an  engineer  is 
indispensible  if  the  transition  curve  is  of  considerable 
length  as  may  occur  when  it  has  been  used  for  the 
purposes  of  securing  a  better  fitting  to  the  ground  in 
location,  irrespective  of  its  use  as  an  adjustment  curve. 
This  latter  contingency  can  usually  be  avoided  by 
introducing  a  circular  curve  of  intermediate  radius 
between  those  to  be  joined,  and  uniting  this  intermediate 
curve  with  the  principal  ones  by  means  of  the  proper 
transition  curves  based  on  the  assumed  values  entering 
the  constant  P.  For  the  above  reasons,  it  is  believed 
that  the  use  of  the  transition  curve  for  other  purposes 
than  adjustment  should  not  be  encouraged.  With  these 
conditions  adhered  to,  a  table  can  be  prepared  giving  the 
ordinates  at  regular  intervals  on  the  long  chord  or  along 
the  length  of  the  transition  curve. 

The  beginning  and  end  of  the  transition  curve 
should  be  marked  by  permanent  points. 


A  few  useful  problems  follow  in  the  application 
of  the  transition  curve. 


PROBLEM  I. 


TO  FIND  THE  SEMI-TANGENTS. 

5  21.  Given  a  circular  curve  whose  radius  =  ru\  the 
intersection  angle  =  /;  the  semi-tangent  =  7",  to  unite 
it  with  tangents  by  means  of  transition  curves  whose 
lengths  are  Lt  and  L  and  offsets  are  /  and  /respectively. 


Fig5 


CASE  i. 
When  f,  >  f  (by  Fig.  5),  if  /<  90°, 


=  B...A  +  Acl-alcl  +  btdt  - 
rf  +  r-/cot/+/cose/.  (i) 


THE   SPIRAL.  37 

If   /       90°, 

Bd  =  BA  4-  Ac  +  ab  +  bd  = 

.rf/  4-  /'-/(-  cot  /)  +/cose  /  ;  (2) 

Bd  =  -rf/  +  T  +/  cot  /  +/cose  /  ;  (3) 

or  in  general  calling, 

B,,,  d,  or  Bd  -  t,;  t,  =  Xf,  +  T  T  £  cot  I  +  f  cose  I,    (4) 

the  +  sign  being  used  when   /     90°  and   the  -  sign 

When  /  <  90°. 

B,  dt  =  B,  A,  +  ^/y  ct  +  c,  b.  -  d.g,  - 

xi  -f  T  +/  cose  /-/cot  /;  (5) 

7^  </  =  /^  ^y  -f  Alt  c  4-  ^  +  dg  - 

.rf+  r  +/  cose  /4-/  cot/  (6) 

In  general  calling, 

By  d,  or  B,  d  =  t  =  xf  +  T  +  t  cose  I     f  cot  I,       (7) 
using  +  when  /     90°  and  -  when  /  '  90°. 

CASE  2. 
If  f  =  0  in  equation  4,  we  have 

t,  =  T  4-  1  ^se  I,      t  =  Xf  4-  T  -f  f,  cot   I.       (8) 

CASE  3. 
If  f,  =  f  in  equation  4, 


//  =  *f/  +  T+f,  (cose  /  ;  cot  /), 

/  =  ^4-  r4-/(cose/  :-  cot/).  (9) 


38  TRANSITION  CURVE. 

If  1  <  90°,  the  last  term  of  /,  is  /  (cose  /-  cot  /)  and  by 
trigonometry  (see  Chauvenet's,  p.  35), 

T  T         i        cos  7      i  -  cos  / 

cose/-  cot  /=    .     .--.  —  f  -  —  :  —  j—  =  tan  1AI  .\ 
sin  1     sin  1  sin  / 

t,  =  Xf  +  T+f,  tan  ^7or  t,  =  .rf/  +  (rn+  /)  tan  #/;  (10) 
similarly,  if  /  >  90°, 

A     I  ^     T  I   +  COS  / 

cose  /  +  cot  /  —  —  -  —  =  —  : 
sin  / 

but  if  7  :  •  90°, 

cos  7  =  -  cos  7 

7-1  <.     7-  T   ~  COS  I 

cose  7+  cot  7=       .  -    .    , 
sin  7 

which   we   have  seen  =  tan  ]^1.      Hence  when  f,  =  f, 

t,  =  t  =  Xf/  +T+  f,  tan  %\  =  xf/  +(!•„  +  f,)tanKI  (") 

is   a   general    equation    whether  7   be    greater    or   less 
than  90°. 

Kx  AMPLE  19.     Given  /.  =  210  ;     r  =  818.8  ;    f  =  2.23  ;     /  -^=  40°; 
OG,   =  104.93  ;  to  find  £  =  Fig.  (5): 

f       />/  -  '/       (/•//       /)  tan  ^7  -  (104.93       821.03)  .36397  =  403.75. 

With  the  same  data 

TO  FIND  THE  EXTERNAL  SECANT. 

CASE  i. 
When  f7     f, 


e.d,  -  (r//  +  f)secA/Dd,-r,,.  (12) 


If  f  -  0, 


THE  SPIRAL. 
CASE  2. 


e,,  b,  =  P,,  sec  A,,  D  b,  -  P,,  . 

CASE  3. 
f,  =  f ,  e/  d,  -  (P,, 


39 


(13) 


(14) 


PROBLEM  II. 

TO  FIND  THE  LOCATION  OF  THE  OFFSET  "f". 

%  22.  Given  two  curves  with  radii  rt  and  rllt  a 
distance  Dt  Dtl  —  d  joining  the  points  Dt  and  Din  also 
the  angles  BD,Dtl  =  p  and  CDllD,  =  et  to  find  the 
points  A,  and  Alt  at  which  a  line  drawn  through  the 
centers  rt  and  rn  at  ^  and  C  will  cut  the  curves  At  D, 
and  AH  Dtl. 


From  Fig.  6  we  have 

FC  =  h  —  ^sinp-ry/siu  (180  -  (/>  -|-  0) ),  (i) 


4o  TRANSITION  CURVE. 

and  since  BC  —  rt  -  (rlt  4-ft\ 

FC_  h  _*/ ship- r,,  sin  (180- (/o+0)_  . 

BC~r,-(rtl+f)  ~  rt-(rtt+f)~ 

DtlCD,  =  2  =  i8o-(p  +  0),  *-S  =  w  =/?,,C  .4,,;    (3) 

whence 

r,,u  =  AltDM  (4) 

in  which  o>  is  an  arc. 

If  2  >  >1>,  then  the  point  ^4,,  lies  between  Dn  and  Z>, 
and  the  distance  AtlDz  is  measured  from  D '.,  towards 
-#//  to  .4,,. 

If  ^  >  2,  the  point  ^f,,  lies  beyond  Dlt  and  the 
distance  An  Dtl  is  measured  from  Dtl  to  ^/;  whence  ^/y  is 
established.  On  a  perpendicular  to  a  tangent  at  An  and 
from  Alt  lay  offy;  and  establish  At. 

When  /  is  small,  the  direction  of  the  radial  line 
can  be  estimated  near  enough.  The  method  of  fixing 
Bt  and  Bin  in  Fig.  2,  has  already  been  indicated. 
If  rt  —  co,  Dt  A,  is  tangent.  Fig.  7  applies. 


EXAMPLE  20.    r,  =  1000  ;   rn  -  600  ;  rf  =  300  ;  p  =  70°;   ».=  96°; 
/•-50': 

1st  to  find  A  =  rfsinp    -  ;-/y  sin  (180  -  (p  -  0»)  =  300  X  .93969  - 
600  y:  .24192  =  136.75  =  FC. 

k  136.75 

2nd   to   find    sm  *  =    ^  _  ^    ,  y;)    =  -jjg^-.^--        = 


3rd  to  find  180  —  [70°  \-  96°J  =  14°=  2;  *  —  2  =  w  =  23°—  14°=  9°; 

w  reduced  to  arc  =  -^-^77  =  .1571. 
57.29 

ria  =  600  X  .1571  =  94.26  =  A,,  /),,  ,  and  since  *  :  2,  -47/  /^/y  is 
measured  from  D,,  towards  Dj. 

If  2  =  D^C  D^t  then  2  i.-1*  and  AltD^  would  be  measured 
from  7>3  towards  Du  to  establish  point  ^4y/;  A,  is  on  a  perpendicular 
to  a  tangent  to  the  curve  with  radius  r,,  at  A,,. 


THE  SPIRATv.  41 


PROBLEM  III. 


3 

I  2¥.  To  unit  two  circular  curves  turning  in  the 
same  direction,  whose  centers  are  a  fixed  distance  apart, 
by  the  introduction  of  a  third  circular  curve  of  an 
intermediate  radius  and  to  unit  the  same  with  the  main 
curves  by  the  use  of  transition  curves  : 


Given  two  circular  curves  with  radii  r,  and  r.,, 
respectively,  whose  centers  are  apart  a  distance  A  C  —  b  = 
ri  ~  (rz  +.///)  i  atl(i  which  are  separated  from  each  other  a 
distance  Dt  Dn  =  ftl.  It  is  desired  to  introduce  between 
them  a  third  curve  with  radius  rlt  ,  less  than  r,  and 
greater  than  r.^  and  to  join  the  curve  with  radius  rlt  with 
those  having  radii  r,  and  r..  by  means  of  transition 
curves  : 

By  the  figure  AD,  =  AA,  =  ABt  =  rt  =  AC  +  CD  + 
DltD,  =  b  +  r,  +  fu,  whence 


(i) 


42  TRANSITION   CURVE. 

AB  =  c  =  r,-(r,,+/,);  (2) 

BC  =  a  =  r,,-(r3+/3);  (3) 

a,  b  and  c  form  the  sides  of  a  triangle  ABC  in  which 


Any  angle  as  ^4  may  be  found  by  the  formulae, 


Ver  A  =  2      ~fc~      iu  which  s  =  --  (^+^+^  ;  (6) 
Sin^  =  -~-sin  ^4;  (7) 

Sin  C  =  —  sin  -4  ;  (8) 

Reducing  each  of  the  above  angles  to  arc  as  indicated  in 
another  part  of  this  book  we  have, 

A,  D,  =  r,A  •  A.  i  Dtl  -  r;J  0;  AltA,,  =  rlt  (iSo-B)  (9) 
the  arc 


-  "" 

in  the  same  way, 


THE  SPIRAL.  43 

(13) 


A,B,  =  r,ft;     AtlBn  =  r,,ft,;    A3  B,  =  r,,j83; 

AiBt  =  r8flt!  (H) 

J3tl£3=ritd;     d  =  (iSo-£)  -  (ft,  +  j83);  (15) 

£,/>,  -  r,  J.  +  r,  ft  -  r,  (4  +  ft);  (16) 


EXAMPLE  21.  Given  r;=2865;  r/y=1146;  r3=  716.3;  ///=20; 
/>=  171900;  A-  90;  L3  =  90  .•.//=/3=  .17  to  join  the  circular  curves 
^Z>y  and  #4  />>„  by  Prob.  III.  First  finding  a,  b  and  c  by  (1-3) 
we  have  by  (6)  ver  A  =  3°44'  ;  arc  A  =  .06516;  by  (7)  (180—  ^)=18°48'; 
arc  (180  -  £)=.32811;  by  (8)  C-  15°06';  arc  C=.  26054;  ^/  A=  2865  >< 
.06516  =  186.68;  A±D,,=  716.3  X-26054  =  186.77;  A,,A-A  =1146X.3281= 

376'00;  ^=  - 


=  .0157;  ^4;^=^  ltBtl=A*  £3=A±  £4  =  ±o.Q;  5=18048'-(2015'+20150 
=14°20';  arc  6  =.25017;  Bn  JS3  =  1146X.  25017  =286.7;  B,  #,=2865  (.06516 
f.  0157)  =231.67;  B±  Dn  =716.3  (.26054+  .06282)  =231.62. 

If  we  make  f,  =  o  and  r,  =  rtl  ,  then  c  —  o,  6'  =  o, 
ftl  and  ^It  —  Q^b  =  a  and  is  coincident  with  it,  J.  =  180  -  B 
and  the  problem  reduces  to  uniting  AtlA^  with  Dlt  B±  by 
means  of  the  transition  curve.  If  rn  =  r,  ,  c  =  o,  C  =  o, 
in  which  case  f,  and  f3  =  o'a  is  equal  to  and  coincides 
with  b  and  J.  —  180  -  It,  equations  for  vers  =  o.  The 
problem  reduces  to  uniting  curves  B,  Dt  and  B±  Dtl  by 

means  of  a  transition  curve  whose  length  L2  = 


If  f3  and/7  =  o  while  r, ,  rlt  and  r3  retain  their  values, 
the  transition  curves  disappear  and  the  curve  with  radius 
rlt  compounds  at  A4  and  A,  with  the  curves  having 
radii  r,  and  r3. 

180-^  is  the  central  angle  of  Atl  A3;  (18) 

C  =  that  of  DHA±\        A  =  that  of  A,  D,        (19) 


*For  determining  large  central  angles  of  /3, ,  (3,, ,  /3;Jj  and  j8,t  the 
method  of  §16  is  to  be  preferred. 


44 


TRANSITION  CURVE. 


PROBLEM  IV. 


I  22.  Given  two  curves  turning  in  the  same  direc- 
tion, whose  centers  are  a  distance  apart  =  DtDtl  and 
whose  radii  are  r,  and  rn.  To  fix  the  position  of  a  tan- 
gent A,  Ain  and  connect  it  with  the  circular  curves  An  C, 
and  A±Clt  by  means  of  transition  curves  having  a  fixed 


Re- 


value of  P  (Fig.  9).  Lst  C,  Ctl  be  a  line  joining  any  two 
points  C,  and  Cn  of  the  circular  curves  with  radii  rt  and  rn\ 
D,  D,,  the  known  distance  between  the  centers  of  the 
curves  r,  and  rn.  Measure  the  angles  Dt  C,  Ctt  and 
C,  C,,  £>„.  By  traverse  we  find  the  angles  CID,DII 
and  C^D^D,.  If  from  Dt  and  Dtl  we  let  fall  perpen- 
diculars to  any  imaginary  tangent  passing  through  the 
origins  of  the  transition  curves,  by  the  conditions  of  the 
problem  the  length  of  these  perpendiculars  will  be: 


D.  A,  =  r,  +f,  and  Dtt  Ani  =  rtl  +/„  , 
and  any  distance  D,  Kt  =  (r,  +/)  -  (r,,  +  //), 


(i) 


THE   SPIRAI,.  45 

and  if  we  denote  the  distance  D,  Dtl  by  D  4,  then 


G7,  Z>,  ^//  =  difference  of  Gr,  Z>,  D,,  and  0.  Ct  D,  Atl  reduced 
to  arc  and  multiplied  by  r,  =  the  distance  C,  Alt  to 
establish  Atl  ,  from  Afl  lay  off  ft  normal  to  the  circle  and 
establish  At.  In  the  same  way  Clt  D/t  A±  =  the 
difference  between  180°  0  and  Ctl  D,,  D,  . 

GffD^A^  multiplied  by  rtl  —  arc  distance  G,,An  from 
Glt  to  A4  to  establish  A.it  from  which  lay  off  flt  and 
establish  Atll\  then  a  line  through  A,  A  in  is  the  required 
tangent. 

The  distance  AtO,  to  the  origin  0,  =  x,  -  r,  sin  at 
and  the  distance  Ain  Ou  =  xn  -  r,,  sin  an\  whence  the 
distance 

0,  On  =  D,  Dtl  sin  6  -  [(*„  -  rtl  sin  ay/)+(.r/  -  r,  sin  a;)].  (3) 
If  we  wish  the  distance  00  =  zero,  make 

I)i  Dn  —  \_(xn  -  rn  sin  al^-\-(.rl  -  r,  sin  ay)]  cose  ^.     (4) 

This  gives  the  shortest  possible  distance  between  the 
centers  of  the  circular  curves  where  the  transition  curves 
are  introduced.  The  transition  curves  may  have  different 
values  of  P  provided  A,  Ot  -\-  Ain  Ou  —  or  <  than  A,AIU. 
If  the  curves  are  reversed,  Dt  Kn  =  (r,  +//)  4-  (rtl  +///) 
and  we  find  the  value  of  9  by  completing  the  traverse 
0,  CHI  Dtll  D,  Cr 


EXPLANATION  OF  THE  TABLES. 


BY  RECTANGULAR  CO-ORDINATES. 


$  25.  Tables  I  to  IX  give  values  for  laying  out  the 
transition  curve  by  the  method  of  rectangular  co-ordinates, 
They  are  equally  applicable  for  uniting  a  tangent  with  a 
circular  curve,  or  curves  having  different  radii,  by  means 
of  the  transition  curve.  Llt  and  L,  may  be  taken  sepa- 
rately from  the  same  column  as  also  may  a,,  and  a, ,  and 
their  difference  will  be  the  value  of  L  and  0  for  the 
length  and  central  angle  respectively.  The  several 
ordinates,  JT,  yt  xf,  yf,  are  laid  off  from  B,  or  Bn  as 
origin  (Fig.  2)  with  arc  B,At  or  BtlAlt  as  axis, — the 
same  as  if  B,  or  Bn  were  written  for  A  in  Fig.  3,  and  the 
successive  stations  were  B,  C}  D,  etc.  and  B, ,  Ct ,  D,  ,  etc., 
successive  points  x,  x, ,  xni ,  etc.  with  corresponding 
y,y, ,  ytl ,  etc. ,  values  normal  to  the  curved  axis  A,  B,  Atl  Blt 
in  the  same  manner  as  if  A,  U,  were  tangent.  Values  of 
d  are  dropped,  as  in  no  case  in  these  tables  do  they  reach 
j\y  of  a  foot.  Table  II  is  extended  as  an  example  to  develop 
these  differences  to  an  amount  worthy  of  consideration. 
P  and  v  are  taken  of  such  values  as  to  avoid  introducing 
fractions  in  L.  L  is  supposed  to  be  measured  on  the  curve, 
but  since  the  chords  are  generally  quite  short,  the 
sum  of  their  lengths  is  but  little  less  than  that  of  the 
curve,  hence  no  allowance  is  made  for  the  length  of  the 
curve  being  in  excess  of  the  sum  of  the  lengths  of  the 
chords. 


EXPLANATION   OF  THE  TABLES.  47 

EXAMPLE  I.  Given  I,  -  120  ;  r,  -  1432.5;  rn  =  716.3  (Table  II), 
then  at/  —  a,  =s  9°  36'  —  2°  24'  =  7°  12'  =  <£  with  E  as  origin.  At  the 
end  of  the  first  chord  length  from  E  towards  /  we  have  x  =  30', 
y  =  .03  =  co-ordinates  for  F;  x,  —  60,  y,  =  .21  =  co-ordinates  for  G; 
Xll  =  90,  y,,  =  .70  =  co-ordinates  for  H;  a; 3  —  119.98,  r3  =  1.67  = 
co-ordinates  for  /;  xt  =  60;  yf  =  .21;  /=  42. 

The  method  of  laying  out  these  co-ordinates  is  shown  in  Fig.  3, 
in  which  the  origin  A  corresponds  to  the  point  E  in  this  example, 
and  A,  D,  becomes  a  curved  axis  with  a  radius  of  1.432.5. 

If  the  transition  curve  were  laid  off  from  B,,  An  as  axis  and 
£,,  as  origin,  the  above  values  x  &c  and  y  &c  would  be  just  the 
same  except  they  would  be  laid  off  from  the  convex  side  of  A,,  Rn 
instead  of  from  the  concave  side,  as  was  the  case  with  A,  B,  as  axis. 
If  the  curvature  of  the  circular  curve  is  of  fractional  degree  the  value 
of  /and  the  last  values  of  x,  y  and  a  wilt  have  to  be  computed  by 
the  formulae  at  the  head  of  the  respective  column  in  the  table. 


TABLE  la  TO  Va  BY  DEFLEXION. 

£  26.  Given  a  tangent  at  any  point  of  a  transition 
curve  as  D  to  locate  any  other  points  as  A,  By  C,  E,  F,  G. 
As  in  the  case  of  the  Tables  for  rectangular  co-ordinates 
they  are  equally  applicable  to  locating  the  points  of  the 
transition  curve  uniting  a  tangent  to  the  circular  curve  or 
circular  curves  of  different  radii  with  each  other.  (Fig.  4) 

EXAMPLE  2.  I^et  L,,  —Lt=*L=  120  be  the  length  of  the 
transition  curve;  r/  =  1910;  rn  =955,  the  radii  of  the  circular 
curves  to  be  united  by  L  by  deflections  from  a  tangent  at  D,  where 
the  curvature  corresponds  to  r,.  The  tangent  will  be  common  to 
the  circle  and  transition  curve  whose  rate  of  curvature  "Z>"  (Table 
II)  =-3°.  DE,  EF,  FG  etc.,  being  chords  of  the  transition  curve 
each  — 30  feet.  The  tangent  at  D  is  a  tangent  to  the  circular  arc 

with  r/       1910.     Then  by  the  formula  w  =  D  A  L  -f-  -0 • ,    in     which 

o 

/;  -  3°;  A  =  .?-  ;   DE^L  =  ^  feet ;     J  =  0°  3' ;    3  X.3'  X  30  +  0°03'  = 
1U  o 

w  =  27'  +  03'  =  0°  30'  - the  deflection  from  tangent  at  D  to  locate 
points.  If  we  want  to  locate  the  point  F  from  D,  then  L  =  60  ; 
D  =  3  A  =  13'  ;  •'  =  12' ;  o>  =  54  ^  12  =  66'  =  1°  06'  =  deflection  from 


48          EXPLANATION  OF  THK  TABLES. 

tangent  at  D  to  locate  F  with  measurement  from  D  to  E  thence 
E  to  F.  If  we  wish  to  locate  the  points  C,  B  and  A  from  a  tangent 
at  D,  then  the  deflection  for  any  point  C  '  =  the  deflection  for  30 
feet  for  a  3  degree  curve  minus  the  deflection  for  the  transition 
cnrve  from  A  to  £,  or 

o>=/?AL--£=3X3X30-  4  =27'-  3'=  24'. 

o  o 

•SO' 
The  deflection  for  the  poiut  ^  =  3  X-3'X  60  -  --—  =  0°54'-0°12'=0°42. 

o 

If  we  have  run  the  curve  from  A  to  D  and  changed  the  instrument 
to  D  in  order  to  place  the  line  of  sight  tangent  to  D,  take  a  back 
sight  on  A  and  deflect  0°  54'  we  have  a  tangent  at  Z>.  To  facilitate 
the  use  of  the  tables  it  is  best  to  set  the  vernier  at  0°  54'  and  set  the 
telescope  on  line  AD,  turn  the  vernier  to  O  and  continue  deflection 
as  tabulated,  reading  downward  from  Z>,  locating  the  points 
E,  F,  G  and  C.  If  the  curve  is  being  run  from  D  towards  A  then 
set  the  vernier  at  the  angle  indicated  for  any  angle  G,  when  back 
sight  is  on  G  from  D  deflect  from  zero  and  continue  to  deflect  the 
angles  tabulated  in  succession,  reading  up  the  column  from  D  to 
locate  C,  S  and  A.  The  degree  of  the  curvature  at  the  instrument 
point  controls  the  deflections  either  way.  The  above  explanation 
enables  us  to  run  the  transition  curve  from  the  point  of  greatest 
radii  to  that  of  its  least  radius,  and  vice  versa. 

To  use  the  tables  when  the  beginning  or  end  of  the 
transition  curve  merges  into  a  circular  curve  of  a  frac- 
tional degree,  as  joining  a  2^  degree  curve  with  a 
6  degree  curve,  or  a  2  degree  curve  with  a  5^  degree 
curve  ;  compute 


for  15  feet,  then 

2.5  X  .3'  X  15  +  oo'.ys  =  o°  12', 

which   locates  a  point  D  at  3  degrees  curvature  ;  then 
from  D,  with  back  sight  on  C  -\-  15  ; 

2.5  X  .3'  X  15  +  oi'.so  =  o°  i2'.75 


UNIVERSITY) 


EXPLANATION  OF  THE  TKSffSS^'  49 

will  be  the  deflection  to  fix  a  tangent  at  D  from  which 
point  proceed  with  deflections  down  the  D  column.  If 
the  last  chord  FG  is  one-half  L,  the  deflections  of  the 
table  can  be  followed  up  to  F  to  which  point  the  instru- 
ment can  be  moved  and  tangent  established  from  which 
deflect 

w  =  DA  —  +  oo'.75  =  5  X  .3'  X  15  +  oo'.75  =  o°  23'. 

If  it  is  desired  to  deflect  the  whole  angle  from  D  to 
G  when  F  G  =  one-half  L  when  Lf  =  the  length  from 
DtoF, 

ZG  =  /:,  +  —-,  then 

2 


r  =  2296  ;      -T-?  X  57°  18'  =  5  =  o°  19'; 


*>  =  D  A      Lf  +      -       +  -        X  57D  18'  = 

3  X  .3'  X  75  +  19'  -  67'  +  19'  =  i°  26'  . 

Any  other  fraction  of  L  than  I/L  may  be  used  in  the 
formulae. 


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55 


A  =  deflection  angle  per  foot  for  1°  circular  curve. 

5  =  -^—deflection  angle  from  tangent  at  A  towards  G  for  points 

o 

of  spiral. 

D  =  degree  or  rate  of  curvature  at  position  of  instrument. 

o>  =  deflection  from  tangent  at  any  point  of  spiral  to  any  other 
point  of  spiral.    Then 

o>  =  D  A  L  +  -4-=  D  A  L  ±  -^-57°  18' 

3  6  r 

-f  sign  for  running  towards  G  —  sign  towards  A. 
The  instrument  point  is  the  origin  of/,  and  x.  x  conceived  to  to  be 
laid  off  on  the  circular  curve  passing  through  instrument  point  as  axis 


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6o 


PROBLEM  V. 


To  introduce  the  transition  curve  in  alignment 
where  simple  curves  have  been  run.  In  Fig  10,  suppose 
ABC,  a  simple  circular  curve,  to  have  been  run  tangent 
to  the  line  O  Gtl  at  A  with  radius  rl  =  DlA=D,B;  it  is 
desired  to  introduce  a  spiral  whose  greatest  curvature 
has  a  radius  rn  -  Dn  Atl ,  Dtl  Bn  ,  DttB<rt.  From  the 
tables  or  by  computation  we  have  m  depending  on  the 
value  of  rlt  and  Ltl ;  with  given  values  of  m,  rlt  and  r,  we 
have  from  the  Figure 


,      ~~* 

A  Dt  =  A,  A,,+  An  Dtl-\-  Dlt  D,  cos  Dtt  D,K\ 

rt  =  m,  4-  rn  +  (rt  -  r,,)  cos  0 . '. 
m  =  r,  -  rn  -  (r,  -  rn  )  cos  0 
m  -  r,  -  rtl  (i  -  cos  0)  =  (r,  -  rlt]  ver  0 
m 


ver0  = 


r,  (f>  =  AB;    rn  0  =  Atl  B 


AB  =  the  distance  to  measure  from  PC  to  locate 
B\  BAn  the  distance  to  measure  from  B  on  BAlt  to 
locate  An  .  From  m  and  rtl  we  have 


24  m  rlt  ,  squaring,  4  r?  a,/2  —  24  m  rlt 
6w  .  L 


2  rtl  a  =  L  =  l 


rM  (0/y  -  a/y)  =  .5^,  ;  (r  - 
The  rate  of  curvature  of  rtt  should  not  be  more  than 
from  i°  to  2°  greater  than  that  of  r,  (when  possible) 
for  curves  of  a  curvature  less  than  10°;  2°  to  3°  difference, 
for  10°  to  15°  rate  of  curvature;  3°  to  5°  difference  for  15° 
to  20°  rate  of  curvature. 


